单轴对称工字形单悬伸梁和双跨连续梁整体稳定性能研究
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摘要
弯扭屈曲是钢梁的重要的破坏模式之一,不同的截面形式、边界条件以及荷载形式对梁的失稳模式和稳定承载力有很大影响。从十九世纪末开始,各国学者针对各种形式的梁进行了试验研究和理论分析,有关各种条件下单跨梁的整体稳定性能的研究已经很成熟,各国规范中用于计算单跨梁的整体稳定承载力公式也基本一致。
由于悬伸梁和连续梁屈曲问题的复杂性,目前还没有简便的方法来计算其弯扭屈曲荷载。国外学者对双轴对称截面的连续梁进行过试验研究,国内尚未查阅到相关的试验资料。现今学者们提出用于计算悬伸梁和连续梁的整体稳定承载力计算方法多针对双轴对称截面,而且荷载作用点高度和截面不对称性对整体稳定性能的影响考虑的不够全面。因此,对单轴对称工字形悬伸梁和连续梁的整体稳定性能及其承载力的计算方法还需进行深入的研究。
本课题采用试验与理论分析相结合的方法,对单轴对称工字形单悬伸梁和双跨连续梁的整体稳定性能进行了较为系统的研究,并据此提出了可考虑截面形式、荷载作用位置的临界弯矩和极限弯矩的简化计算公式。
考虑到国内缺少悬伸梁和连续梁的试验数据,本文对5根单轴对称工字形单悬伸梁和3根单轴对称工字形双跨连续梁进行了集中荷载作用在上翼缘时的整体稳定承载力试验研究。试验结果表明:当单悬伸梁控制稳定的部分在简支段,且大部分上翼缘受压时,相同条件下,上翼缘加强截面的极限承载力明显高于下翼缘加强截面。当自由端荷载位置不变,简支段的集中荷载越靠近悬伸段支座,悬伸段对简支段的约束越大,梁的极限承载力也随之增大;对于双跨连续梁,当控制梁段跨度不变时,随着约束梁段的跨度逐渐增大,梁段间的相互约束逐渐减弱,梁的极限承载力降低。并且当约束梁段上作用荷载为零时,其对控制梁段的约束最小。在进行梁的整体稳定试验时,可利用Southwell法获得较精确的构件弹性临界荷载,从而可以利用有限的试件得到更多的试验数据。
采用截面法对焊接单轴对称工字形截面的残余应力分布进行了测量。根据实测值建立了接近试验情况的单轴对称工字形截面残余应力分布模型,并用FORTRAN语言编写了能自动生成初应力文件的程序。利用ANSYS进行非线性屈曲分析时读入初应力文件,计算得到的极限承载力与试验结果吻合较好。经分析发现不考虑残余应力的有限元结果与试验结果相差较大。
在试验的基础上,分别对单轴对称工字形单悬伸梁和双跨连续梁进行了大量的参数分析。通过研究截面形式、截面尺寸、梁跨度、相邻梁段跨度比及荷载作用点高度等参数对极限弯矩的影响发现:对于单悬伸梁,当集中荷载作用在下翼缘,悬伸段对简支段的约束随悬伸长度与简支段跨度比η的增大或简支段与悬伸段荷载之比R的减小而增强;当集中荷载作用在上翼缘,R值较大时,悬伸段对简支段的约束随η增大而增强,而当R值较小时,约束随η增大反而减弱。单悬伸梁受均布荷载时,梁段间的约束随η增大而增强,与相同条件下集中荷载作用时相比,R对极限弯矩的影响不如η大。对于双跨连续梁,约束梁段的跨度越大,梁段间的相互约束越弱。
以有限元分析结果为依据,通过对集中荷载(或均布荷载)作用下单轴对称工字形单悬伸梁、双跨连续梁临界弯矩与相同条件下单跨简支梁临界弯矩之比Mc/Mce(或Md/Mde)的研究,发现在其它条件一定时,Mc/Mce(或Md/Mde)随控制梁段的扭转刚度系数K1的变化呈抛物线变化。由此回归得到了单轴对称工字形单悬伸梁和双跨连续梁临界弯矩的计算公式。通过对弹塑性弯扭屈曲模拟结果的回归得到了极限弯矩与临界弯矩的关系式。
开展了单、双轴对称工字形双跨、单跨梁在焊接残余应力影响下整体稳定性能的探索性研究。分析了荷载形式、荷载作用高度、截面不对称性、梁跨度及翼缘残余拉、压应力面积之比对梁极限承载力的影响,并提出了考虑残余应力梁的极限弯矩与折算长细比的关系式。通过分析发现,当受压翼缘残余拉、压应力面积之比大于1时,残余应力会提高出平面相对长细比较大的梁的极限承载力,其提高程度不容忽视。这与本文的试验结果完全吻合。
The flexural-torsional buckling is an important failure mode of the steel beams. The buckling modes and stability capacities are significantly influenced by the variations in the section shape, condition of support and the type of loading. From the end of nineteen century, experimental investigation and theory analysis on all kinds of beams are being carried by many scholars. The researches on the overall stability capacity of the single span beams under various conditions are already very mature, and the formulas for calculating this capacity are also similar to in the design codes among different countries.
Until now, the simple and convenient methods of calculating the buckling loads of the overhanging beams and continuous beams have not been presented, because this problem is considerably complex. Some tests were conducted on the double-symmetric I-section continuous beams by the foreign scholars, but the related experiment data have not been found in the domestic. Actually the proposed formulas for calculating the overall stability capacity mostly focus on the double-symmetric I-section beams, and in which the influence of the loading position and the monosymmetry section constant have not been fully accounted for. Therefore the overall stability behavior and the methods of calculating the overall load-carrying capacity of the monosymmetric I-section overhanging beams and continuous beams still need to be further investigated.
Tests and theoretical analysis on the overall stability behavior of the monosymmetric I-section overhanging beams and two-span continuous beams were systematically conducted in this paper. Based on above, the simplified formulas for calculating the critical moment and ultimate moment have been proposed, in which the section shapes and the loading positions are all considered.
Because of lack of the experimental data about the overhanging beams and continuous beams in the domestic, a total of 8 tests on the overall stability capacity of monosymmetric I-section beams, including 5 overhanging beams and 3 continuous beams, were performed under concentrated load applied on the top flange. As indicated from the experimental results: when the dominative stability portion of the overhanging beam is at the simply supported segment and greater portion of the top flange is compressed, the ultimate load of the beam with the enhanced flange at the top is greater than the beam with the enhanced flange at the bottom under the same conditions. When the position of the load acting at the free end of the beam is constant, the restraint to the simply segment by the overhanging segment is gradually enhanced with the position of the concentrated load acting at the simply segment closing to the support of the overhanging segment, and the ultimate load-carrying capacity of the overhanging beam is also gradually increases. For the continuous beams, if the span of the critical segment keeps invariable, the decrease in the buckling load is greater and greater as the span of the restraining segment gradually increases, due to the interactions between the segments becoming gradually lower. The interactions between the segments are the lowest when the load acting at the restraining segment is zero. The fairly accurate critical load can be obtained by using the Southwell plot method, consequently the more experimental data can be received from the elastic tests by using the limited specimens.
The residual stresses of the welded monosymmetric I-section are measured by the sectioning method. According to the measured results, the simplified residual stress patterns of the monosymmetric I-section which are close to the test distribution have been proposed. The programs which can automatically create the initial stress file have been also compiled using FORTRAN. The initial stress file can be inputted in the non-linear buckling analysis by using ANSYS, the calculated ultimate loads fit together well with the test results. It is observed that the results of finite element analysis (FEA) in which the residual stresses are not considered are quite different with the test results.
Based on the tests, large amounts of finite element parametric analysis were performed for the monosymmetric I-section overhanging beams and two-span continuous beams. The stability behaviors of these members were investigated by changing the section shapes, the section dimensions, the span lengths, the ratio of the span lengths, the load position and so on. It is observed that, the restraining actions between adjacent segments of the overhanging beams under concentrated loads acting at the bottom flange increase as the increase ofη(the ratio of overhanging length to span length of the simply segment) or the decrease of R (the ratio of the loads between the simply segment and the overhanging segment); the restraining actions between adjacent segments of the overhanging beams under concentrated loads acting at the top flange increase as the increase ofηwhen R is bigger, whereas the restraining actions between adjacent segments decrease as the increase ofηwhen R is smaller. The restraining actions between adjacent segments of the overhanging beams under the distributed loads increase as the increase ofη. The effect of the R is not as much asηwhen the overhanging beams carry distributed loads comparing with the concentrated loads. For the two-span continuous beams, the restraining actions between adjacent segments gradually decrease as the increase of the span length of the restraining segment.
Taking the FEA results as the reference, the investigation about the ratio of the critical moments between the monosymmetric I-section overhang beams or two-span continuous beams under concentrated loads (or distributed loads) and the simply supported beams under the same conditions Mc/Mce (or Md/Mde) were conducted. It is found that the interrelation between the ratios of critical moments, Mc/Mce (or Md/Mde), and the coefficients of torsional stiffness of the dominative segment, K1, agree with the parabolic pattern, therefore the formulas for the critical moment of the monosymmetric I-section overhanging beams and two-span continuous beams can be obtained. The relation equations between the ultimate moment and critical moment are also obtained by using the regression analysis for the results of inelastically flexural-torsional buckling simulation.
The studies about the effect of the residual stress on the overall stability capacity of the monosymmetric and double-symmetric I-section two-span and single-span beams were carried out. The influence of the loading type, the loading position, the monosymmetry section constant and the ratio of the distribution area between the tensile and the compress residual stress in the flange were all considered. The relation equations between the ultimate moment allowing for the residual stress and normalized slenderness have been proposed. It is shown that the ultimate bearing capacities of the beam with larger normalized slenderness may increase by the residual stress when the ratio of the distribution area between the tensile and the compress residual stress in the flange is greater than 1, and the increment should not be neglected. It is quite close to the test results of this paper.
由于悬伸梁和连续梁屈曲问题的复杂性,目前还没有简便的方法来计算其弯扭屈曲荷载。国外学者对双轴对称截面的连续梁进行过试验研究,国内尚未查阅到相关的试验资料。现今学者们提出用于计算悬伸梁和连续梁的整体稳定承载力计算方法多针对双轴对称截面,而且荷载作用点高度和截面不对称性对整体稳定性能的影响考虑的不够全面。因此,对单轴对称工字形悬伸梁和连续梁的整体稳定性能及其承载力的计算方法还需进行深入的研究。
本课题采用试验与理论分析相结合的方法,对单轴对称工字形单悬伸梁和双跨连续梁的整体稳定性能进行了较为系统的研究,并据此提出了可考虑截面形式、荷载作用位置的临界弯矩和极限弯矩的简化计算公式。
考虑到国内缺少悬伸梁和连续梁的试验数据,本文对5根单轴对称工字形单悬伸梁和3根单轴对称工字形双跨连续梁进行了集中荷载作用在上翼缘时的整体稳定承载力试验研究。试验结果表明:当单悬伸梁控制稳定的部分在简支段,且大部分上翼缘受压时,相同条件下,上翼缘加强截面的极限承载力明显高于下翼缘加强截面。当自由端荷载位置不变,简支段的集中荷载越靠近悬伸段支座,悬伸段对简支段的约束越大,梁的极限承载力也随之增大;对于双跨连续梁,当控制梁段跨度不变时,随着约束梁段的跨度逐渐增大,梁段间的相互约束逐渐减弱,梁的极限承载力降低。并且当约束梁段上作用荷载为零时,其对控制梁段的约束最小。在进行梁的整体稳定试验时,可利用Southwell法获得较精确的构件弹性临界荷载,从而可以利用有限的试件得到更多的试验数据。
采用截面法对焊接单轴对称工字形截面的残余应力分布进行了测量。根据实测值建立了接近试验情况的单轴对称工字形截面残余应力分布模型,并用FORTRAN语言编写了能自动生成初应力文件的程序。利用ANSYS进行非线性屈曲分析时读入初应力文件,计算得到的极限承载力与试验结果吻合较好。经分析发现不考虑残余应力的有限元结果与试验结果相差较大。
在试验的基础上,分别对单轴对称工字形单悬伸梁和双跨连续梁进行了大量的参数分析。通过研究截面形式、截面尺寸、梁跨度、相邻梁段跨度比及荷载作用点高度等参数对极限弯矩的影响发现:对于单悬伸梁,当集中荷载作用在下翼缘,悬伸段对简支段的约束随悬伸长度与简支段跨度比η的增大或简支段与悬伸段荷载之比R的减小而增强;当集中荷载作用在上翼缘,R值较大时,悬伸段对简支段的约束随η增大而增强,而当R值较小时,约束随η增大反而减弱。单悬伸梁受均布荷载时,梁段间的约束随η增大而增强,与相同条件下集中荷载作用时相比,R对极限弯矩的影响不如η大。对于双跨连续梁,约束梁段的跨度越大,梁段间的相互约束越弱。
以有限元分析结果为依据,通过对集中荷载(或均布荷载)作用下单轴对称工字形单悬伸梁、双跨连续梁临界弯矩与相同条件下单跨简支梁临界弯矩之比Mc/Mce(或Md/Mde)的研究,发现在其它条件一定时,Mc/Mce(或Md/Mde)随控制梁段的扭转刚度系数K1的变化呈抛物线变化。由此回归得到了单轴对称工字形单悬伸梁和双跨连续梁临界弯矩的计算公式。通过对弹塑性弯扭屈曲模拟结果的回归得到了极限弯矩与临界弯矩的关系式。
开展了单、双轴对称工字形双跨、单跨梁在焊接残余应力影响下整体稳定性能的探索性研究。分析了荷载形式、荷载作用高度、截面不对称性、梁跨度及翼缘残余拉、压应力面积之比对梁极限承载力的影响,并提出了考虑残余应力梁的极限弯矩与折算长细比的关系式。通过分析发现,当受压翼缘残余拉、压应力面积之比大于1时,残余应力会提高出平面相对长细比较大的梁的极限承载力,其提高程度不容忽视。这与本文的试验结果完全吻合。
The flexural-torsional buckling is an important failure mode of the steel beams. The buckling modes and stability capacities are significantly influenced by the variations in the section shape, condition of support and the type of loading. From the end of nineteen century, experimental investigation and theory analysis on all kinds of beams are being carried by many scholars. The researches on the overall stability capacity of the single span beams under various conditions are already very mature, and the formulas for calculating this capacity are also similar to in the design codes among different countries.
Until now, the simple and convenient methods of calculating the buckling loads of the overhanging beams and continuous beams have not been presented, because this problem is considerably complex. Some tests were conducted on the double-symmetric I-section continuous beams by the foreign scholars, but the related experiment data have not been found in the domestic. Actually the proposed formulas for calculating the overall stability capacity mostly focus on the double-symmetric I-section beams, and in which the influence of the loading position and the monosymmetry section constant have not been fully accounted for. Therefore the overall stability behavior and the methods of calculating the overall load-carrying capacity of the monosymmetric I-section overhanging beams and continuous beams still need to be further investigated.
Tests and theoretical analysis on the overall stability behavior of the monosymmetric I-section overhanging beams and two-span continuous beams were systematically conducted in this paper. Based on above, the simplified formulas for calculating the critical moment and ultimate moment have been proposed, in which the section shapes and the loading positions are all considered.
Because of lack of the experimental data about the overhanging beams and continuous beams in the domestic, a total of 8 tests on the overall stability capacity of monosymmetric I-section beams, including 5 overhanging beams and 3 continuous beams, were performed under concentrated load applied on the top flange. As indicated from the experimental results: when the dominative stability portion of the overhanging beam is at the simply supported segment and greater portion of the top flange is compressed, the ultimate load of the beam with the enhanced flange at the top is greater than the beam with the enhanced flange at the bottom under the same conditions. When the position of the load acting at the free end of the beam is constant, the restraint to the simply segment by the overhanging segment is gradually enhanced with the position of the concentrated load acting at the simply segment closing to the support of the overhanging segment, and the ultimate load-carrying capacity of the overhanging beam is also gradually increases. For the continuous beams, if the span of the critical segment keeps invariable, the decrease in the buckling load is greater and greater as the span of the restraining segment gradually increases, due to the interactions between the segments becoming gradually lower. The interactions between the segments are the lowest when the load acting at the restraining segment is zero. The fairly accurate critical load can be obtained by using the Southwell plot method, consequently the more experimental data can be received from the elastic tests by using the limited specimens.
The residual stresses of the welded monosymmetric I-section are measured by the sectioning method. According to the measured results, the simplified residual stress patterns of the monosymmetric I-section which are close to the test distribution have been proposed. The programs which can automatically create the initial stress file have been also compiled using FORTRAN. The initial stress file can be inputted in the non-linear buckling analysis by using ANSYS, the calculated ultimate loads fit together well with the test results. It is observed that the results of finite element analysis (FEA) in which the residual stresses are not considered are quite different with the test results.
Based on the tests, large amounts of finite element parametric analysis were performed for the monosymmetric I-section overhanging beams and two-span continuous beams. The stability behaviors of these members were investigated by changing the section shapes, the section dimensions, the span lengths, the ratio of the span lengths, the load position and so on. It is observed that, the restraining actions between adjacent segments of the overhanging beams under concentrated loads acting at the bottom flange increase as the increase ofη(the ratio of overhanging length to span length of the simply segment) or the decrease of R (the ratio of the loads between the simply segment and the overhanging segment); the restraining actions between adjacent segments of the overhanging beams under concentrated loads acting at the top flange increase as the increase ofηwhen R is bigger, whereas the restraining actions between adjacent segments decrease as the increase ofηwhen R is smaller. The restraining actions between adjacent segments of the overhanging beams under the distributed loads increase as the increase ofη. The effect of the R is not as much asηwhen the overhanging beams carry distributed loads comparing with the concentrated loads. For the two-span continuous beams, the restraining actions between adjacent segments gradually decrease as the increase of the span length of the restraining segment.
Taking the FEA results as the reference, the investigation about the ratio of the critical moments between the monosymmetric I-section overhang beams or two-span continuous beams under concentrated loads (or distributed loads) and the simply supported beams under the same conditions Mc/Mce (or Md/Mde) were conducted. It is found that the interrelation between the ratios of critical moments, Mc/Mce (or Md/Mde), and the coefficients of torsional stiffness of the dominative segment, K1, agree with the parabolic pattern, therefore the formulas for the critical moment of the monosymmetric I-section overhanging beams and two-span continuous beams can be obtained. The relation equations between the ultimate moment and critical moment are also obtained by using the regression analysis for the results of inelastically flexural-torsional buckling simulation.
The studies about the effect of the residual stress on the overall stability capacity of the monosymmetric and double-symmetric I-section two-span and single-span beams were carried out. The influence of the loading type, the loading position, the monosymmetry section constant and the ratio of the distribution area between the tensile and the compress residual stress in the flange were all considered. The relation equations between the ultimate moment allowing for the residual stress and normalized slenderness have been proposed. It is shown that the ultimate bearing capacities of the beam with larger normalized slenderness may increase by the residual stress when the ratio of the distribution area between the tensile and the compress residual stress in the flange is greater than 1, and the increment should not be neglected. It is quite close to the test results of this paper.
引文
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17 F. Mohri, A. Brouki and J. C. Roth. Theoretical and Numerical Stability Analyses of Unrestrained, Mono-Symmetric Thin-Walled Beams. Journal of Constructional Steel Research. 2003,59(1): 63-90
18 T. M. Roberts, R. Narayanan. Strength of Laterally Unrestrained Monosymmetric Beams. Thin-Walled Structures. 1988,6: 305-319
19 M. A. Bradford. Effect of Secondary Warping on Lateral Buckling. Engineering Structures. 1989,11(2): 112-118
20 T. M. Roberts, M. Benchiha. Lateral Instability of Monosymmetric Beams with Initial Curvature. Thin-Walled Structures. 1987,5: 111-123
21 Y. -L. Pi, N. S. Trahair. Prebuckling Deflections and Lateral Buckling. Ⅰ: Theory. Journal of Structural Engineering. 1992,118(11): 2949-2966
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23 N. S. Trahair. Laterally Unsupported Beams. Engineering Structures. 1996, 18(10): 759-768
24 J. Valentino, N. S. Trahair. Torsional Restraint Against Elastic Lateral Buckling. Journal of Structural Engineering. 1998,124(10): 1217-1225
25 B. Gosowski. Spatial Buckling of Thin-Walled Steel-Construction Beam-Columns with Discrete Bracings. Journal of Constructional Steel Research. 1999,52(3): 293-317
26 B. Gosowski. Spatial Stability of Braced Thin-Walled Members of Steel Structures. Journal of Constructional Steel Research. 2003,59(7): 839-865
27 J. Valentino, Y. -L. Pi and N. S. Trahair. Inelastic Buckling of Steel Beams with Central Torsional Restraints. Journal of Structural Engineering. 1997,123(9): 1180-1186
28 T. V. Galambos. Inelastic Lateral Buckling of Beams. Journal of the Structural Division. 1963,89(ST5): 217-242
29 N. S. Trahair, S. Kitipornchai. Buckling of Inelastic I-Beams Under Uniform Moment. Journal of the Structural Division. 1972,98(ST11): 2551-2566
30 D. A. Nethercot. Residual Stress and Their Influence Upon the Lateral Buckling of Rolled Steel Beams. The Structural Engineer. 1974,52(3): 89-96
31 D. A. Nethercot. Inelastic Buckling of Steel Beams under Non-uniform Moment. The Structural Engineer. 1975,53(2): 73-78
32 H. Yoshida, K. Maegawa. Lateral Instability of I-Beams with Imperfections. Journal of Structural Engineering. 1984,110(8): 1875-1892
33 D. A. Nethercot, N. S. Trahair. Inelastic Lateral Buckling of Determinate Beams. Journal of the Structural Division. 1976,102(ST4): 701-717
34 D. A. Nethercot. Inelastic Buckling of Monosymmetric I-Beams. Journal of the Structural Division. 1973,99(ST7): 1696-1701
35 S. Kitipornchai, A. D. Wong-Chung. Inelastic Buckling of Welded Monosymmetric I-Beams. Journal of Structural Engineering. 1987,113(4): 740-756
36 S. Bild, G. Chen and N. S. Trahair. Out-of-Plane Strengths of Steel Beams. Journal of Structural Engineering. 1992,118(8): 1987-2003
37 Y. -L. Pi, N. S. Trahair. Inelastic Bending and Torsion of Steel I-Beams. Journal of Structural Engineering. 1994,120(12): 3397-3417
38 C. J. Earls. On the Inelastic Failure of High Strength Steel I-Shaped Beams. Journal of Constructional Steel Research. 1999,49 (1): 1-24
39 N. S. Trahair, G. J. Hancock. Steel Member Strength by Inelastic Lateral Buckling. Journal of Structural Engineering. 2004,130(1): 64-69
40 H. A. Sawyer. Post-Elastic Behavior of Wide-Flange Steel Beams. Journal ofthe Structural Division. 1961,87(ST8): 43-71
41 Y. Fukumoto, Y. Itoh. Statistical Study of Experiments on Welded Beams. Journal of the Structural Division. 1981,107(ST1): 89-103
42 S. Kitipornchai, N. S. Trahair. Buckling of Inelastic I-Beams Under Moment Gradient. Journal of the Structural Division. 1975,101(ST5): 991-1004
43 S. Kitipornchai, N. S. Trahair. Inelastic Buckling of Simply Supported Steel I-Beams. Journal of the Structural Division. 1975,101(ST7): 1333-1347
44 P. F. Dux, S. Kitipornchai. Inelastic Beam Buckling Experiments. Journal of Constructional Steel Research. 1983,3(1): 3-9
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52 M. A. Bradford, N. S. Trahair. Distortional Buckling of I-Beams. Journal of the Structural Division. 1981,107(ST2): 355-370
53 M. A. Bradford. Distortional Buckling of Monosymmetric I-Beams. Journal of Constructional Steel Research. 1985,5: 123-136
54 M. A. Bradford, S. W. Waters. Distortional Instability of Fabricated Monosymmetric I-Beams. Computers and structures. 1988,29(4): 715-724
55 M. A. Bradford. Inelastic Distortional Buckling of I-Beams. Computers and structures. 1986,24(6): 923-933
56 M. A. Bradford. Buckling Strength of Deformable Monosymmetric I-Beams. Engineering Structures. 1988,10(7): 167-173
57 M. A. Bradford. Buckling of Elastically Restrained Beams with Web Distortions. Thin-Walled Structures. 1988,6(3): 287-304
58 M. A. Bradford. Distortional Buckling Strength of Elastically Restrained Monosymmetric I-Beams. Thin-Walled Structures. 1990,9: 339-350
59 M. A. Bradford. Lateral-Distortional Buckling of Steel I-Section Members. Journal of Constructional Steel Research. 1992,23(1): 97-116
60 M. Ma, O. Hughes. Lateral Distortional Buckling of Monosymmetric I-Beams under Distributed Vertical Load. Thin-Walled Structures. 1996,26(2): 123-145
61 O. Hughes, M. Ma. Lateral Distortional Buckling of Monosymmetric Beams under Point Load. Journal of Structural Engineering. 1996,122(10): 1022-1029
62 M. A. Bradford. Distortional Buckling of Elastically Restrained Cantilevers. Journal of Constructional Steel Research. 1998,47(1-2): 3-18
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17 F. Mohri, A. Brouki and J. C. Roth. Theoretical and Numerical Stability Analyses of Unrestrained, Mono-Symmetric Thin-Walled Beams. Journal of Constructional Steel Research. 2003,59(1): 63-90
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23 N. S. Trahair. Laterally Unsupported Beams. Engineering Structures. 1996, 18(10): 759-768
24 J. Valentino, N. S. Trahair. Torsional Restraint Against Elastic Lateral Buckling. Journal of Structural Engineering. 1998,124(10): 1217-1225
25 B. Gosowski. Spatial Buckling of Thin-Walled Steel-Construction Beam-Columns with Discrete Bracings. Journal of Constructional Steel Research. 1999,52(3): 293-317
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27 J. Valentino, Y. -L. Pi and N. S. Trahair. Inelastic Buckling of Steel Beams with Central Torsional Restraints. Journal of Structural Engineering. 1997,123(9): 1180-1186
28 T. V. Galambos. Inelastic Lateral Buckling of Beams. Journal of the Structural Division. 1963,89(ST5): 217-242
29 N. S. Trahair, S. Kitipornchai. Buckling of Inelastic I-Beams Under Uniform Moment. Journal of the Structural Division. 1972,98(ST11): 2551-2566
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32 H. Yoshida, K. Maegawa. Lateral Instability of I-Beams with Imperfections. Journal of Structural Engineering. 1984,110(8): 1875-1892
33 D. A. Nethercot, N. S. Trahair. Inelastic Lateral Buckling of Determinate Beams. Journal of the Structural Division. 1976,102(ST4): 701-717
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35 S. Kitipornchai, A. D. Wong-Chung. Inelastic Buckling of Welded Monosymmetric I-Beams. Journal of Structural Engineering. 1987,113(4): 740-756
36 S. Bild, G. Chen and N. S. Trahair. Out-of-Plane Strengths of Steel Beams. Journal of Structural Engineering. 1992,118(8): 1987-2003
37 Y. -L. Pi, N. S. Trahair. Inelastic Bending and Torsion of Steel I-Beams. Journal of Structural Engineering. 1994,120(12): 3397-3417
38 C. J. Earls. On the Inelastic Failure of High Strength Steel I-Shaped Beams. Journal of Constructional Steel Research. 1999,49 (1): 1-24
39 N. S. Trahair, G. J. Hancock. Steel Member Strength by Inelastic Lateral Buckling. Journal of Structural Engineering. 2004,130(1): 64-69
40 H. A. Sawyer. Post-Elastic Behavior of Wide-Flange Steel Beams. Journal ofthe Structural Division. 1961,87(ST8): 43-71
41 Y. Fukumoto, Y. Itoh. Statistical Study of Experiments on Welded Beams. Journal of the Structural Division. 1981,107(ST1): 89-103
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